Contents
Context
Homotopy theory
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Model category theory
model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
for ∞-groupoids
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
-Limits and colimits
limits and colimits
1-Categorical
-
limit and colimit
-
limits and colimits by example
-
commutativity of limits and colimits
-
small limit
-
filtered colimit
-
sifted colimit
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connected limit, wide pullback
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preserved limit, reflected limit, created limit
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product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
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finite limit
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Kan extension
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weighted limit
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end and coend
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fibered limit
2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Idea
The simplicial bar construction is a way of building a cofibrant resolution of a diagram, suitable for computing homotopy colimits in simplicial model categories.
Definition
Let be a cocomplete symmetric monoidal closed category and let be a small -category. We define a simplicial object for each pair of objects in by
with the obvious face and degeneracy operators induced by the composition and identities of . Note that is a -functor .
Let be a tensored -category. The two-sided simplicial bar construction of a -diagram weighted by a -functor is a simplicial object equipped with isomorphisms
that are natural in and -natural in , where the integral sign on the right hand side denotes a -end.
Now suppose a cosimplicial object is given, so that we may define the realisation? of a simplicial object in a -category as the weighted colimit . The two-sided bar construction of weighted by is then defined to be the geometric realisation of the two-sided simplicial bar construction:
Properties
Proposition
If is a tensored -category and has coproducts for small families of objects, then has two-sided simplicial bar constructions for all diagrams and weights. If is moreover -cocomplete, then also has two-sided bar constructions.
Proposition
If is a tensored -category and has coproducts for small families of objects, then:
- For each -diagram , the -functor has a right -adjoint.
If is also cotensored, then:
- For each weight , the -functor has a right -adjoint.
Proposition
Let be the realisation of . If is a cotensored -cocomplete -category, then there are isomorphisms
that are -natural in .
The next theorem is essentially a version of the Fubini theorem for coends.
Theorem
Let , , and be -functors. If is a cotensored -cocomplete -category, then we have the following isomorphism:
Applications
As stated in the introduction, the two-sided simplicial bar construction can be used to compute homotopy colimits.
Let us write for the representable -functor and for the diagram .
Theorem
Let be a simplicial model category and let be a small category.
- For each diagram , there is a weak equivalence that is natural in and .
- The functor preserves weak equivalences between diagrams of the form where is a diagram such that each is a cofibrant object in .
- The functor , where is a cofibrant replacement functor, is a left derived functor for , i.e. it computes homotopy colimits.
References